To give an informative but non-exact picture, you can first imagine a charge q that has a lot of discrete energy levels around. If you increase the amount of the charge, these levels would get closer to each other (and when you make two atoms close together, it is like that you have a bigger system with larger amount of charges and closer energy levels) so that in the limit you gather a lot of atoms, the levels get very close and make a continuous energy band.

1 far away enough that they do not affect each other, that is why we regard them as two alone systems.
2 "splitting" is referring to the energy,
3 of course different as the picture implies due to Pauli's exclusion principle,